Question

In $\Delta TUV$, how do you express $\cos T$ in terms of $t, u, v$?

Answer 1

Since we have been told to express the terms in $\cos T$, we will simply try to use the standard cosine formula. While using this take into consideration the angle in the general formula. Try to match the sides t, u, v and angle given while making the necessary correction for the given questions in the standard cosine formula.

Complete step by step answer:
We have been given the triangle $\Delta TUV$. The vertices of this triangle are T, U, V respectively with sides t, u, v respectively. The standard cosine rule for triangle with vertices A, B, C is given by
${a^2} = {b^2} + {c^2} - 2cb\cos A$
If we had to find $\cos C$ the formula would be
${c^2} = {a^2} + {b^2} - 2ab\cos C$
Since we have to express $\cos T$ in terms of t, u, v , we will use the standard cosine rule for triangles with vertices T, U, V which is given by
${t^2} = {u^2} + {v^2} - 2uv\cos T$
Now rearranging the above equation we get
$2uv\cos T = {t^2} - {u^2} - {v^2}$
solving it further to get all terms in $\cos T$
$\therefore\cos T = \dfrac{{{t^2} - {u^2} - {v^2}}}{{2uv}}$

Hence we have expressed $\cos T$ can be expressed as $\dfrac{{{t^2} - {u^2} - {v^2}}}{{2uv}}$.

Note: Since the main aim of the question above was to express in $\cos T$, we have used the standard cosine formula. Apart from this the important part of the solution is the rearrangement of the formula according to the given conditions and our requirements. With the help of this rule we can find the length of the triangles or can find the angles between the two sides of the triangle.